Integral Equations without a Unique Solution can be made Useful for Solving some Plane Harmonic Problems†

摘要

Using Green's third identity an integral equation for a two dimensional harmonic problem is derived. For a particular exceptional geometry the integral equation does not have a unique solution but by applying Green's third identify a supplementary integral condition is derived. When the integral equation and the integral condition are solved simultaneously we obtain always a unique solution. The procedure is demonstrated by some numerical examples.